Computational Methods Using a Bayesian Hierarchical Generalized Linear Model

Abstract

Abstract Suppose we observe Y 1, …, YN , where Yi has the exponential density f(yi |θi ) = exp{ϕi ([yiθi – b(θi )]}c(yi, ϕi ). The parameters of interest are not the canonical parameters θi but the means μi = b′(θi ). In the usual generalized linear model (GLM) setup, suppose the means μ1, …, μN are believed to satisfy a specific p-dimensional GLM g(μi ) = xT iβ, where the link function g and the regression coefficients {xi } are known and the regression vector β is unknown. Two problems of interest are the assessment of the goodness of fit of the GLM and the estimation of the means μi . The approach to these problems is by the use of a Bayesian two-stage prior distribution, a generalization of a model used by Lindley and Smith (1972) in the normal mean-estimation problem. At the first stage of the model, we assign independent conjugate distributions to θ1, …, θN , where the prior means of the μi satisfy the GLM. There are p + 1 unknown hyperparameters in this specification, the elements of the regression...